1. Field of the Invention
This invention relates generally to the field of geophysical prospecting. More particularly, the invention relates to the field of interpolating or regularizing traces in seismic data.
2. Description of the Related Art
In the oil and gas industry, geophysical prospecting is commonly used to aid in the search for and evaluation of subterranean formations. Geophysical prospecting techniques yield knowledge of the subsurface structure of the earth, which is useful for finding and extracting valuable mineral resources, particularly hydrocarbon deposits such as oil and natural gas. A well-known technique of geophysical prospecting is a seismic survey. In a land-based seismic survey, a seismic signal is generated on or near the earth's surface and then travels downward into the subsurface of the earth. In a marine seismic survey, the seismic signal may also travel downward through a body of water overlying the subsurface of the earth. Seismic energy sources are used to generate the seismic signal which, after propagating into the earth, is at least partially reflected by subsurface seismic reflectors. Such seismic reflectors typically are interfaces between subterranean formations having different elastic properties, specifically sound wave velocity and rock density, which lead to differences in acoustic impedance at the interfaces. The reflected seismic energy is detected by seismic sensors (also called seismic receivers) at or near the surface of the earth, in an overlying body of water, or at known depths in boreholes and recorded.
The appropriate seismic sources for generating the seismic signal in land seismic surveys may include explosives or vibrators. Marine seismic surveys typically employ a submerged seismic source towed by a ship and periodically activated to generate an acoustic wavefield. The seismic source generating the wavefield may be of several types, including a small explosive charge, an electric spark or arc, a marine vibrator, and, typically, a gun. The seismic source gun may be a water gun, a vapor gun, and, most typically, an air gun. Typically, a marine seismic source consists not of a single source element, but of a spatially-distributed array of source elements. This arrangement is particularly true for air guns, currently the most common form of marine seismic source.
The appropriate types of seismic sensors typically include particle velocity sensors, particularly in land surveys, and water pressure sensors, particularly in marine surveys. Sometimes particle displacement sensors, particle acceleration sensors, or pressure gradient sensors are used in place of or in addition to particle velocity sensors. Particle velocity sensors and water pressure sensors are commonly known in the art as geophones and hydrophones, respectively. Seismic sensors may be deployed by themselves, but are more commonly deployed in sensor arrays. Additionally, pressure sensors and particle velocity sensors may be deployed together in a marine survey, collocated in pairs or pairs of arrays.
In a typical marine seismic survey, a seismic survey vessel travels on the water surface, typically at about 5 knots, and contains seismic acquisition equipment, such as navigation control, seismic source control, seismic sensor control, and recording equipment. The seismic source control equipment causes a seismic source towed in the body of water by the seismic vessel to actuate at selected times. Seismic streamers, also called seismic cables, are elongate cable-like structures towed in the body of water by the seismic survey vessel that tows the seismic source or by another seismic survey ship. Typically, a plurality of seismic streamers are towed behind a seismic vessel. The seismic streamers contain sensors to detect the reflected wavefields initiated by the seismic source and reflected from reflecting interfaces. Conventionally, the seismic streamers contain pressure sensors such as hydrophones, but seismic streamers have been proposed that contain water particle velocity sensors such as geophones or particle acceleration sensors such as accelerometers, in addition to hydrophones. The pressure sensors and particle motion sensors may be deployed in close proximity, collocated in pairs or pairs of arrays along a seismic cable.
The resulting seismic data obtained in performing the survey is processed to yield information relating to the geologic structure and properties of the subterranean formations in the area being surveyed. The processed seismic data is processed for display and analysis of potential hydrocarbon content of these subterranean formations. The goal of seismic data processing is to extract from the seismic data as much information as possible regarding the subterranean formations in order to adequately image the geologic subsurface. In order to identify locations in the Earth's subsurface where there is a probability for finding petroleum accumulations, large sums of money are expended in gathering, processing, and interpreting seismic data. The process of constructing the reflector surfaces defining the subterranean earth layers of interest from the recorded seismic data provides an image of the earth in depth or time.
The image of the structure of the Earth's subsurface is produced in order to enable an interpreter to select locations with the greatest probability of having petroleum accumulations. To verify the presence of petroleum, a well must be drilled. Drilling wells to determine whether petroleum deposits are present or not, is an extremely expensive and time-consuming undertaking. For that reason, there is a continuing need to improve the processing and display of the seismic data, so as to produce an image of the structure of the Earth's subsurface that will improve the ability of an interpreter, whether the interpretation is made by a computer or a human, to assess the probability that an accumulation of petroleum exists at a particular location in the Earth's subsurface.
Two problems can arise in the collection of seismic data, whether on land or in water. The data may be under-sampled (aliased) or the data may be non-uniformly (irregularly) sampled. Physical and economic limitations in a seismic survey often cause seismic data to be acquired either as under-sampled or non-uniformly sampled
Under-sampled data is commonly referred to as aliased data. From data sampling theory, it is desired that no wavelength embedded in the data be shorter than twice the sampling interval. Otherwise, the feature corresponding to the embedded wavelength will be under-resolved and hence distorted due to the aliasing.
Thus, the temporal alias frequency at which aliasing begins, the Nyquist frequency, is
  f  =      1          2      ⁢                          ⁢      Δ      ⁢                          ⁢      t      in the frequency-wavenumber (f-k) domain. The spatial alias frequency at which aliasing begins, the Nyquist wavenumber, is
  k  =      π          Δ      ⁢                          ⁢      x      in the spatial coordinates. Here Δt is the sample-time interval in milliseconds and Δx is the station spacing in preferred units, such as meters. Thus, a large sampling interval in the time-space (t-x) domain corresponds to a small Nyquist frequency and Nyquist wave number in the corresponding f-k domain.
Recording the seismic data with much larger sampling intervals than ideally desired can lead to harmful effects in subsequent data processing. However, collecting data on a finer sampling interval during the seismic survey significantly adds to the cost of seismic data acquisition, particularly in the case of 3D surveying. So, instead, missing data can be approximated from the acquired data. Thus, some benefit must be sought by interpolating or extrapolating spatially aliased seismic data.
Interpolation of traces in unaliased, uniformly sampled seismic data is straightforward. The interpolation can be performed, for example, by convolution with a sinc filter in the spatial domain or by extending the Nyquist wavenumber of a band-limited signal through zero padding in the Fourier domain. However, this easier trace interpolation assumes that the interpolation is done with orthogonal basis functions. The energy of the signal in the data leaks to all the other frequencies when the trace interpolations are performed on an irregularly sampled grid. The energy leakage is caused by the irregularities of sampling and boundary effects.
Thus, non-uniformly sampled data needs to be regularized to an orthogonal (regular) basis grid. Three-dimensional seismic data regularization requires generating seismic traces at locations where the acquisition at the source and receiver positions did not take place during the seismic survey. In other words, seismic traces from the acquired data on an irregular grid are interpolated or extrapolated to a regular grid.
Marine seismic data in particular are usually irregularly and sparsely sampled along the spatial directions for many reasons, including cable feathering, obstacle avoidance, editing of bad traces, and economics. However, regularly sampled data are required for several seismic processing applications, including 3D surface-related multiple elimination and 3D wave equation migration. The best way to obtain 3D regularly sampled data is to acquire more data, with more redundancy in the crossline direction and with a wider azimuth range, but this is expensive and difficult to achieve because of current marine acquisition technology. Therefore, data regularization becomes an important processing tool.
Thus, a need exists for a method for interpolating traces in seismic data that is both under-sampled and non-uniformly sampled. In particular, a need exists for a method of trace interpolation that attenuates the energy leakage due to irregular sampling in aliased seismic data.